Pre-Algebra Examples

$| 2 x - 5 | = | 3 - 2 x |$

Step 1

Rewrite the equation as $| 3 - 2 x | = | 2 x - 5 |$ .

$| 3 - 2 x | = | 2 x - 5 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$3 - 2 x = 2 x - 5$

$3 - 2 x = - (2 x - 5)$

$- (3 - 2 x) = 2 x - 5$

$- (3 - 2 x) = - (2 x - 5)$

Step 3

After simplifying, there are only two unique equations to be solved.

$3 - 2 x = 2 x - 5$

$3 - 2 x = - (2 x - 5)$

Step 4

Solve $3 - 2 x = 2 x - 5$ for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $2 x$ from both sides of the equation.

$3 - 2 x - 2 x = - 5$

Step 4.1.2

Subtract $2 x$ from $- 2 x$ .

$3 - 4 x = - 5$

Step 4.2

Move all terms not containing $x$ to the right side of the equation.

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Step 4.2.1

Subtract $3$ from both sides of the equation.

$- 4 x = - 5 - 3$

Step 4.2.2

Subtract $3$ from $- 5$ .

$- 4 x = - 8$

Step 4.3

Divide each term in $- 4 x = - 8$ by $- 4$ and simplify.

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Step 4.3.1

Divide each term in $- 4 x = - 8$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 8}{- 4}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 4$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 8}{- 4}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 4}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $- 8$ by $- 4$ .

$x = 2$

Step 5

Solve $3 - 2 x = - (2 x - 5)$ for $x$ .

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Step 5.1

Simplify $- (2 x - 5)$ .

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Step 5.1.1

Rewrite.

$3 - 2 x = 0 + 0 - (2 x - 5)$

Step 5.1.2

Simplify by adding zeros.

$3 - 2 x = - (2 x - 5)$

Step 5.1.3

Apply the distributive property.

$3 - 2 x = - (2 x) - - 5$

Step 5.1.4

Multiply.

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Step 5.1.4.1

Multiply $2$ by $- 1$ .

$3 - 2 x = - 2 x - - 5$

Step 5.1.4.2

Multiply $- 1$ by $- 5$ .

$3 - 2 x = - 2 x + 5$

Step 5.2

Move all terms containing $x$ to the left side of the equation.

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Step 5.2.1

Add $2 x$ to both sides of the equation.

$3 - 2 x + 2 x = 5$

Step 5.2.2

Combine the opposite terms in $3 - 2 x + 2 x$ .

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Step 5.2.2.1

Add $- 2 x$ and $2 x$ .

$3 + 0 = 5$

Step 5.2.2.2

Add $3$ and $0$ .

$3 = 5$

Step 5.3

Since $3 \neq 5$ , there are no solutions.

No solution

Step 6

List all of the solutions.

$x = 2$

Step 7

Verify each of the solutions by substituting them into $| 2 x - 5 | = | 3 - 2 x |$ and solving.

$x = 2$